Chapter 11.3, Problem 5E

To determine

Find the area under the curve for $y=-x^2+4$ from x = − 2 to x = 2

Answer to Problem 5E

The approximate area under the curve is 11 sq units.

Explanation of Solution

Given:

Find the area under the curve for $y=-x^2+4$ from x = − 2 to x = 2

Concept Used:

It is like sum of areas of 4 rectangles

The graph:

  

Calculation:

It is like sum of areas of 4 rectangles.

We find the area under the curve $y=-x^2+4$ from $x=-2\text{to}x=2$ ,

According to the graph there are four rectangles.

First rectangle from $x=-2\text{to}x=-1$

Second rectangle from $x=-1\text{to}x=0$

Third rectangle from $x=0\text{to}x=1$

Fourth rectangle from $x=1\text{to}x=2$

Width of each rectangle is 1 unit.

We use the midpoint of each rectangle.

Now find the height of each rectangle,

So height of first rectangle would be $x=-1.5\Rightarrow -{(-1.5)}^2+4=1.75$

Height of second rectangle would be $x=-0.5\Rightarrow -{(-0.5)}^2+4=3.75$

Height of third rectangle would be $x=0.5\Rightarrow -{(0.5)}^2+4=3.75$

Height of fourth rectangle would be $x=1.5\Rightarrow -{(1.5)}^2+4=1.75$

The area of 1^st, 2^nd, 3^rd and 4^th rectangles are: $1.75$ , $3.75$ , $3.75$ , $1.75$ sq unit.

So the approximate area under the curve = sum of all 4 rectangles

So the approximate area under the curve: $1.75+3.75+3.75+1.75=11$ sq unit.

Thus, the approximate area under the curve is 11 sq units.